Splitting Madsen-Tillmann spectra II. The Steinberg idempotents and Whitehead conjecture

TL;DR

This paper demonstrates a splitting of certain Madsen-Tillmann spectra at prime 2, revealing new relationships between these spectra and classical classifying spaces, and confirming specific homotopy equivalences.

Contribution

It establishes a splitting of the spectrum (n) off the Madsen-Tillmann spectrum MTO(n) at prime 2, extending previous results and clarifying the structure for n=2.

Findings

(n) splits off MTO(n) at p=2.

For n=2, MTO(2) is homotopy equivalent to BSO(3)_+  (2).

The splitting aligns with classical decompositions of related spectra.

Abstract

We show that, at the prime $p=2$, the spectrum $\Sigma^{-n}D(n)$ splits off the Madsen-Tillmann spectrum $MTO(n)=BO(n)^{-\gamma_n}$ which is compatible with the classic splitting of $M(n)$ off $BO(n)_+$. For $n=2$, together with our previous splitting result on Madsen-Tillmann spectra, this shows that $MTO(2)$ is homotopy equivalent to $BSO(3)_+\vee\Sigma^{-2}D(2)$.